NOTE: This page contains
many links that go to other sites. Use your 'back' button to return to this
page.
This summary page was written at the end of 2003 to
summarize my collection of pages on this site.
However, the world of magic cubes insists on marching on. I will make only
minor edits to this and my other existing pages.
New advances will be presented in what may be a series of new pages titled
Cube_Update1, Cube_
Update2, etc. 
Introduction
The time has come to write finis to what has become
a labor of love!
In late 2001, I decided to write a page about John
Hendricks new magic hypercube definitions. After some preliminary research into
old ‘perfect’ magic cubes, this page was posted to my magic squares (formerly
Geocities) site on March 10, 2002.
I then decided to do additional research on early magic cubes. The end result,
as they say, is history!
I became hooked on the subject when I discovered the large variety of cubes.
On December 14, 2002, I started a new Web site devoted to
magic cubes. I used the same page design as used on my magic squares site. Also,
as on that site, I have acknowledged the work of known contributors. In order to
make my pages simpler for the casual browser to read, I have not included
details of construction methods (except in a few rare occasions). This
information is better handled on specialized sites.
The original intent was to eventually incorporate it into
my magic squares site, but I have decided to keep it independent. However, on
the top of each page are buttons that will link you directly to the introductory
pages of the three major divisions of that site.
Now, one year later, this site has grown to 38 pages, and
I feel like it is a good time to summarize what has been accomplished.
During this past year (2003) much consolidation of past knowledge and many
advances have been made in the field of magic cubes!
These are brought to your attention with special acknowledgements, in the New
Developments and the Summary Tables toward the end of this page.
I have included lots of links, so you may find that this is a convenient method
of browsing through these cube pages.
Of course, I will continue to maintain these pages, and
add new material when appropriate.
Acknowledgements and Thanks.
What has been accomplished on this site is due in no small
measure to help I have received from many sources.
I thank the many magic square and cube hobbyists and
readers of my pages for suggestions, information, and contribution of new
material. I hope you continue to review my pages periodically, and continue to
offer constructive criticism, suggestions, and new material.
I thank my local library Interlibrary Loans department and
many universities and other institutions for locating and providing me with old
documents.
I also acknowledge with thanks, the information I have garnered from many
Internet web sites.
Recent
Developments
A very special thanks to each of the following
persons. Here are some of the new discoveries they made.
I cannot acknowledge here every person who has contributed
to these pages (although they are credited where their material is located).
However, there is a small group of dedicated magic cube fans, who have become as
enthusiastic about the subject as I have. Throughout my endeavor, they have
provide a constant stream of advise, suggestions, and constructive criticism. In
addition, during 2003 they have made many significant advances in magic cube
knowledge! This has added tremendously to the amount of material I have been
able to post. It also, of course, has made the whole subject of magic cubes,
that much more interesting.
Here I list six friends, who I have known for some years,
including links to their home pages. I will mention their major contributions
and provide links to my relevant pages.
And for the sake of completeness, a bit about my contributions.
Christian Boyer
(France)
Christian seems to have limitless time and energy because he seems to get so
much accomplished. From the start of my project, he has been constantly
available for advice and suggestions. In addition, he has:
 Helped in locating and obtaining old French papers.
Fermat,
Huber,
Violle,
Sauveur, Arnoux
and Leibniz are some examples.
 Made amazing advances in multimagic cubes.
 Discovered an order 9 cube with planar diagonals all
magic, and collaborated with Walter Trump on the
order 5 diagonal cube. He calls them perfect magic (a traditional
definition). I call them diagonal magic, as part of a new
coordinated system of hypercube definitions.
John Hendricks (Canada)
John has been a good friend and collaborator since the
early 1990’s. Of those hobbyists mentioned here, he is the only one that lives
close enough that I can visit with in person. He lives about 3 ½ hours away so
we manage to get together about once a year.
The set of magic hypercube definitions he formulated is
what inspired me to publish a magic cube site. These definitions are
summarized on my index
page but discussed in depth on the Perfect and
Perfect 2 pages. The Inlaid
Magic cubes page describes 4 cubes and a tesseract.
I also have a page in my
magicsquares section that displays a variety of his work.
John is a prolific investigator and writer. He has
published over 50 articles and books on the general subject of magic hypercubes.
Also numerous papers on statistics and miscellaneous mathematics. His
contributions to magic cube knowledge are scattered throughout the various
sections of my sites.
Walter
Trump (Germany)
Walter also seems to have boundless energy. His
contributions to magic cube knowledge include:
 Discussion of 4 types of symmetry in magic cubes. All
are selfsimilar.
 Investigation into Magic cube
groups I, II, and III as well as cubes were all planar arrays are magic
squares of groups I, II, and III. These last cubes do not fit into any order 4
Dudeney classes.
 Discovery of orders 7, 6, and 5 cubes with planar
diagonals all magic.
 Finding 4 simple order 5 magic cubes, with all line
sums (including planar diagonals) magic modulo 2,
10, 31, 62.
 He also found an order 5 simple bordered (it contains
an order 3 central magic cube) magic cube that is diagonal magic modulo 3.
All Trump orders 5 and 7 cubes are unique because none of
the 4 directions have ALL pantriagonals correct.
In fact of the 94 different odd order normal magic cubes in my collection (Dec.
17, 2003), ALL have all pantriagonals correct in at least one direction, except
for the following:
Order 
Total number of normal cubes 
Cubes with all diagonals correct in No
directions 
Author & Type 
3 
4 


5 
37 
8
1 
Trump  Simple
Trump  Diagonal 
7 
24 
1 
Trump  Diagonal 
9 
12 
1 
Boyer  Diagonal 
11 
9 


13 
3 


15 
4 
1 
Heinz  Composite 
17 
1 


Aale de Winkel
(The Netherlands)
Over 6 or so years that I have known Aale, he has been a
great help to me. He seems to have an infinite source of patience in explaining
mathematical procedures that I am having trouble comprehending!
He and I have collaborated in several investigations, most notable
quadrant magic squares and
3_D magic stars.
Aale was a great help to me throughout the 2 years I
worked on this project. He offered suggestions, advised of typographical and
other errors, etc.
In addition, he:
 attempted to find a cube that is NOT magic because no
orthogonal lines are correct BUT all pantriagonals are. He found a cube with
rows and columns all incorrect, with only pillars (and all pantriagonals)
correct! I show it on my Unusual Cubes
page.
 constructed two order 5 magic cubes, simple and
pantriagonal, that are diagonal magic modulo
5.
 supplied me with a number of cubes to help fill in my
collection.
 was a source of inspiration and ideas for all in our
group.
Guenter Stertenbrink (Germany)
Guenter came on the scene only recently (October, 2003),
but provided me with a copy of Planck’s 1905 paper, and reconstructed Planck’s
order 15 perfect magic cube ( Planck had published only
construction details).
Guenter also provided an order 4 pantriagonal magic cube that is a closed
knight tour, and later, his own version
of the order 15 perfect magic cube.
Guenter does not have his own web pages yet.
Mitsutoshi
Nakamura (Japan)
Mitsutoshi also came on the scene recently.
He came to my attention when he quickly filled in some vacant spots in my
summary table. Since then he has created an excellent web site and has produced
a large variety of magic hypercubes.
He established the fact that there are 18
classes of magic tesseracts and has constructed examples of each one(July 2008).
In addition, he:

discovered and constructed an example of a sixth class of magic
cubes, the Pantriagonal Diagonal (Jan. 2005).

has compiled and listed an excellent group of
magic hypercube definitions.

has published (on his site) several proofs
pertaining to magic hypercubes.

created a number of algorithms for
constructing various types and orders of magic hyercubes
Harvey Heinz (Canada)
I include myself in this list because of some additions I
have contributed to magic cube knowledge. I have:
 Collected over 300 cubes (2005) from orders 3 to 17 and
analyzed them by comparing over 15 features.
 Constructed an order 4 sixinone magic cube model and
several composition magic cubes.
 Collaborated with Walter Trump on the Dudeney Group IV
to VI and identified transformations.
 Found that the patterns suggested by G. Arnoux are very
general in ALL magic squares and cubes!
(It seems strange that no one in the 115+ years since Arnoux published this
that this is a source of many magic patterns in virtually every magic square
and cube!)
 Collaborated with Aale de Winkel in the discovery,
investigation, and naming of oddorder Quadrant magic squares
 Defined and provided examples of semipantriagonal
magic cubes.
 Defined the ‘diagonal' magic cube. The name was
suggested by Aale de Winkel.
This cube type fills a hole accidentally left in John Hendricks ‘simple’,
‘pantriagonal’, ‘pandiagonal’, ‘perfect’ list of magic cube classes.
 Helped John Hendricks with the definitions and
publicizing of the new magic cube definitions which now (2005) consists of 6
main classifications.
 Coauthored with John Hendricks The Magic Square
Lexicon: Illustrated (2000).
Two
Summary Tables
In the first of these two tables I will show the first
cube for each order and class that I have actually seen and tested.
The second table lists cubes that are unusual for one reason or another.
In each case I will provide a link to take you to that
cube, complete with reference, if it is on my site.
I have included a footnote for cubes not shown on my site. Note
[6] and higher are for cubes added to this
table after it was first posted.
ADDENDUM: Jan. 2005. This table is now incomplete. A 6th
class of magic cubes has been discovered, by Mitsutoshi Nakamura. It is a
combination Pantriagonal and diagonal magic cube. So far the only known cube is
order 8. It is called a Pantriagonal Diagonal magic cube, or PantriagDiag for
short. More on Definitions and
Update3 pages.
First cube of each class for each order
Order 
Simple Magic 
Pantriagonal
magic 
Diagonal magic
[19] 
Pandiagonal magic

Nasik perfect
magic [18] 
3 
Hugel 1876 
None
possible 
None
possible 
None
possible 
None
possible 
4 
Kurushima
1757 [15] 
Frost
1878 
None
possible 
None
possible 
None
possible 
5 
Hugel
1876 
Hendricks
1972 
Trump/Boyer
2003 
None
possible 
None
possible 
6 
Firth
1889 
Abe
1948 
Trump
2003 
None
possible 
None
possible 
7 
Soni
2001 
Hendricks
1973 
Trump
2003 
Frost
1866 
None
possible 
8 
Andrews
1908 
Frost
1866 
Frankenstein
1875 
Soni
2004 [12] 
Barnard
1888 
9 
Golunski
1984 
Soni
2001 
Boyer
2003 
Soni
2004 [9] 
Planck
1905 [1] 
10 
Planck
1894 
Nakamura 2004 [13] 
Li Wen
1988 [14] 
None
possible [10] 
None
possible [10] 
11 
Soni
2001 
Suzuki
2000? 
Nakamura
2004 [11] 
Soni
2004 [9] 
Barnard
1888 
12 
Poyo
1999 
de Winkel
2003 
Benson & Jacoby
1981 [8] 
None
possible [10] 
None
possible [10] 
13 
Soni
2004 [6] 
Golunski
2003 
Nakamura
2004 [11] 
Soni
2004 [9] 
Liao & associates
1999 
14 
Soni
2003 
Nakamura
2004 [13] 
Benson & Jacoby 1981 [8] 
None
possible [10] 
None
possible [10] 
15 
Heinz
2003 [3] 
Soni
2004 [6] 
Nakamura/Soni
2004 [17] 
Soni
2004 [9] 
Planck
1905 [2] 
16 
Boyer
2003 [4] 
de Winkel
2003 
Nakamura
2004 [16] 
Soni
2004 [12] 
Soni
2003 
17 
Soni
2004 [7] 
Soni
2004 [6] 
Nakamura
2004 [11] 
Soni
2004 [9] 
Arnoux
1887 [5] 
[1] Frost published a Perfect order 9
in 1878, but it was not normal. It used numbers in the series from 1 to 889.
[2] Planck provided instructions only. The cube was constructed by Stertenbrink
in November, 2003.
[3] This cube is shown also in Special Cubes table because it is composite.
[4] This cube is shown also in Special Cubes table because it is bimagic.
[5] The first normal perfect magic cube?
[6] In February, 2004. I generated 3 additional cubes for the above table, using
a program supplied by Abhinav Soni.
[7] I received a Simple (?) order 17 cube from Abhinav Soni on Feb. 11, 2004.
This cube must be classified as 'simple', but actually contains 36 pandiagonal
and 2 simple magic squares.
[8] Benson & Jacoby, Magic Cubes New Recreations, Dover,1981, 0486241408, pp
105115 and pp 116126.
[9] I received these 5 (plus an order 19) pandiagonal magic cubes from Abhinav
Soni on March 9, 2004.
[10] Proved by Stertenbrink and de Winkel . See Pandiagonal Impossibility Proof.
This was previously proved by B. Rosser and
R. J. Walker, Magic Squares: Published papers and Supplement, 1939, a bound
volume at Cornell University, catalogued as QA 165 R82+pt.14.
[11] I received these 3 Diagonal magic cubes from Mitsutoshi Nakamura on March
23, 2004.
These cubes are also unusual. Only 4 planes in each orthogonal are simple magic
squares. All others and all 6 oblique squares
are pandiagonal magic.
NOTE: Nakamura suggested the term 'proper' for cubes that have only the minimum
features required for their class.
These cubes (and those of [7] ) would not be 'proper'! [8], [12], [13], [14],
[16] are some that are proper.
[12] I received these two Pandiagonal cubes from Abhinav Soni on March 29, 2004.
[13] I received these 2 Pantriagonal cubes from Mitsutoshi Nakamura on Apr. 11,
2004.
[14] I downloaded Li Wen's Order 10 cube from Christian Boyer's page. (He (and
some others) still use the old definition of 'Perfect').
[15] From Akira Hiriyama & Gakuho Abe, Researchs in Magic Squares, Osaka
Kyoikutosho, p. 154. (Nakamura email Apr. 18, 2004)
[16] I received this proper Diagonal cube from Mitsutoshi Nakamura on Apr. 18,
2004.
[17] I received this cube from Mitsutoshi Nakamura on Apr. 28, 2004. He reported
that he had help from Abhinav Soni on this one.
That is fitting because the two of them filled the 18 cells in the above table
that were vacant when I posted the table at the beginning of 2004.
Thanks and congratulations Mitsutoshi and Abhinav ! Read more about the
unusual order 15 diagonal cube on Update2.
Mitsutoshi Nakamura is a 40 year old
computer programmer living in Morioka, Japan. He majored in mathematics at
university, but has studied magic cubes only since 2000. He is unmarried and is
an admirer of Yoshihiro Kurushima (? – 1757). See his new Website on magic
cubes.
Abhinav Soni is a graduate student in the Bachelor of Technology degree at the
Indian Institute of Technology in Roorkee, India. His interest in mathematics
led him to write a program to generate magic cubes.
[18] Due to confusion with the term
perfect, Nasik is now the preferred title for a hypercube with all possible line
summing correctly.
See Planck's 1905 enhanced
definition of Frost's 1866 nasik.
[19] Trump now also refers to the diagonal type as strictly magic, again to
lesson the confusion over the term perfect.
Special (unusual)
Cubes
Order

Date 
Constructed by 
Type  Remarks 
3 
1899 
Fourrey 
Not magic. All planes sum to 126 Method copied from
Sauver's 1710 order 5 
4 
1640 
Fermat 
Not magic. No triagonals correct. 8 simple magic
squares 
4 
1838 
Violle 
Not magic. All planes sum to 520.All diagonals sum to
130 
5 
1710 
Sauveur 
Not magic. All planes sum to 1575. All 4 triagonals
O.K. 
4 
1996 
Brown/Cass 
The 'projection' cube. Binary digits in cells project decimal numbers on
the surface. 
8 
1917 
Dudeney 
Not magic. The 6 X 8 surface cells form a closed
knight tour 
4 
1918 
Czepa 
Not magic. Closed knight tour 
4 
2003 
Stertenbrink 
Pantriagonal magic. Closed knight tour. 
4 
2003 
de Winkel 
Not magic unique because only pantriagonals and
pillars are correct. No rows or columns. 
3 
1913 
Sayles 
Multiply magic Constant is 27,000 
4 
1913 
Sayles 
Multiply magic Constant is 57,153,600 
5 
2001 
Trenkler 
Multiply magic Constant is 35,286,451,200 
3 
1977 
Akio Suzuki 
Prime Associated All numbers are prime – S = 3309 
4 
1977 
Gakuko Abe 
Prime Not associated All numbers are prime – S =
4020 
4 
1968 
Les Card 
Not magic – the 4 cells of each line form a 4 digit
reversible prime number. 
4 
2003 
Trump5 
Simple – Semipantriagonal Horizontal planes are
group I 
4 
2003 
Trump6 
Simple – Semipantriagonal Horizontal planes are
group II 
4 
2003 
Trump7 
Simple – Semipantriagonal Horizontal planes are
group III 
4 
1922 
Weidemann2 
Simple  Horizontal planes are group IV 
5 
2003 
de Winkel 
Simple – but perfect magic modulo 5 
5 
2003 
de Winkel 
Pantriagonal – but perfect magic modulo 5 
5 
2003 
Trump6 
Simple – but diagonal magic modulo 2 
5 
2003 
TrumpBordered 
Simple – but diagonal magic modulo 3 – concentric
(contains an order 3 cube in the center) 
5 
2003 
Trump7 
Simple – but diagonal magic modulo 10 
5 
2003 
Trump5 
Simple – but diagonal magic modulo 31 
5 
2003 
Trump3 
Simple – but diagonal magic modulo 62 
6 
1910 
Sayles 
Simple magic. Special feature cubelets 
7 
1922 
Weidemann 
Classed as simple magic because ALL orthogonal planes
are not magic squares, but it does contain 14 pandiagonal and 5 simple
magic squares 
7 
1838 
Violle 
Not magic – but all 21 planar and 6 diagonal planes
sum to 8428 i.e. 7 x 1204. 
8 
1991 
Hendricks 
Simple magic. Inlaid. Contains an order 4
pantriagonal magic cube in the center. 
8 
2003 
Heinz 
The 8 octants are all order 4 pantriagonal, compact, and complete magic
cubes 
9 
2003 
Heinz 
Simple magic. Composite. Consists of 27 order 3 magic
cubes. 
12 
2003 
Heinz 
Simple magic. Composite. Consists of 27 order 4 magic
cubes. 
15 
2003 
Heinz 
Simple magic. Composite. Consists of 27 order 5 magic
cubes. 
4 
2002 
Heinz 
6 in 1 (model). The 64 numbers on each face of each
cell form an order 4 magic cube. 
16 
2003 
Boyer 
Simple magic. This cube is bimagic so when each
number is squared, the cube is still magic. This is the first of Christian
Boyer’s multimagic cubes of different degrees. 
Challenges
Magic cubes  unanswered questions (as of December,
2003).
Pantriagonal magic cubes
Are all normal order 4 pantriagonal
magic cubes either compact or complete (or both)?
No. As of Dec. 15/03 I had 19 pantriagonal order 4 cubes.
12 were both compact and complete, 4 were complete only, 2 were compact only and
Stertenbrink’s (Nov. 9/03) Knight Tour cube was neither!
Compact = Every 2x2 square sums to 130; Complete = Every pantriagonal contains
m/2 complement pairs spaced m/2 apart.
All order 7 pandiagonal magic cubes I examined have at
most, just 1 pandiagonal oblique magic square (the other 5 are simple magic
squares).
Is it possible for more than one of the 6 oblique squares
in an order 7 pandiagonal magic cube to also be pandiagonal magic?
Guenter Stertenbrink (Nov. 29/03) thinks the answer is no, as a result of
counting the number of path directions possible.
The 11 normal pandiagonal magic cubes I have seen (to
Dec./03) are all associated!.
Are there NO pandiagonal magic cubes that are not
associated?
Yes. On Sept. 12/03, Aale de Winkel sent me a normal
pandiagonal magic cube that was not associated.
Guenter Stertenbrink (Nov. 29/03) made a common mistake by stating “All
are NOT associated, because you can move planes from 1 side of the cube to the
other to destroy the association.” But no, shifting planes works with the
equivalent of a pandiagonal magic square, which is a pantriagonal magic cube!
Shifting planes in a pandiagonal magic cube destroys the triagonals.
A pandiagonal magic cube consists of 3m orthogonal
pandiagonal magic squares AND NOT all pantriagonals are correct (or it would be
a perfect cube).
Are all pandiagonal magic cubes order 7? Who will be the
first to construct one of a different order?
Abhinav Soni constructed ones for orders 8, 9, 11, 13, 15, 16, and 17 in March
2004!
Is an order 10 pandiagonal or
perfect cube possible?
Not a normal (consecutive numbers) pandiagonal or nasik
perfect one. This has been proved many times for singly even pandiagonal
magic squares.
Aale de Winkel (Jan. 2004) posted on his encyclopedia site a summary of George
Chen and Guenter Stertenbrink impossibility proofs.
Nasik (Perfect) magic cubes
Corners of all orders 2, 3, 4, 5, 6, 7 and 8 cubes within
all four of the order 8 perfect cubes I have seen sum correctly to 2052.
Is this feature (called compactplus) always present in
order 8 perfect cubes? YES.
In all 4x perfect cubes (x>1)? NO. See March, 2010
update.
Addendum: My Soni perfect order 16 has only corners of subcubes 2, 4, 5, 6, 8,
9, 10, and 16 summing correctly. (HDH Nov. 29/03) so compact_2,5,9, but not
compactplus.
All 4 order 8 perfect cubes also have another feature in
which every pantriagonal contains m/2 complement pairs spaced m/2 apart.
Is this also common to all order 8 perfect cubes? To all
8x perfect cubes? See above update link.
Addendum: The Soni perfect order 16 cube (NOT associated) also has this feature
(it is called complete). (HDH Nov. 29/03)
Addendum2: The Nakamura perfect order 16 cube is
associated and does NOT have this feature, so the answer to the last question is
NO. This cube has corners of subcubes 3, 5, 7, 11, 13, and 15 summing
correctly so is compact_3,5.
Prime number magic cubes:
There has not been much work done with magic cubes
consisting of prime numbers.
What is the smallest possible (i.e. has the smallest
constant) prime number magic cube?
What is the smallest possible magic cube consisting of consecutive primes?
Order 4 magic cube groups:
So far I have seen no magic cubes that correspond to magic
square groups 7 to 12.
When one is found that fit within group 7, 8, 9, or 10, the other three will be
available by swapping planes. Likewise for groups 11 and 12.
Who will be the first to find cubes for some of these
groups?
Walter Trump has found examples of 4 cubes that do not fit
into any of the 12 Dudeney groups. It is not surprising that such cubes exist,
given the fact that cubes are so much more complicated than squares.
Who will be the first to find cubes belonging to
additional groups?
Who will be the first to find order 4 magic cubes with all horizontal planes
that are magic squares of groups 4,or greater?
Miscellaneous cube challenges:
I have not yet seen a combination Add/Multiply magic cube.
Is it possible to construct one?
It is possible to have a number
square where all pandiagonals , but NO rows or columns are correct.
In 2002, such a square was found. It is an
order 4 square with all pandiagonals, but no rows or columns, summing to 34.
Peter D. Loly, A Purely Pandiagonal 4*4
Square..., Journal of Recreational Mathematics, Vol. 31, No. 1, 20022003, pp
2931.
Is it possible to construct a cube with no correct
orthogonal lines, but all pantriagonals are correct?
Is it possible to construct such a cube, but with all pandiagonals correct as
well?
Yes. I received such a cube from Guenter Stertenbrink on
Jan. 4, 2004.
A heterosquare has all line sums different. A subset of
this is the antimagic square, with all line sums different but consecutive.
Who will be the first to construct a heterocube or an
antimagic cube?
On Jan. 9, 2004, I received an order 3 heterocube
from Peter Bartsch that was almost an antimagic cube!
On Jan. 12 and 13, 2004 I received prime number heteromagic cubes from Peter.
See these on update1.
